Mathematics 1210  PRACTICE EXAM III  Fall 2004

1. Calculate the following:

a. lim_{x→0} \frac{tan x − sin x}{x^{2} tan x} b. lim_{x→0} \frac{x^{10/3}}{sin x − x}

c. lim_{x→π/2} \frac{cos x}{x − π/2} d. lim_{x→0} \frac{∫_{0}^{x} (1 − cos t) dt}{x^{3}}

e. lim_{n→∞} \frac{1}{n} ∑_{i=1}^{n} sin \frac{i}{n}

2. Find the general solutions to the following differential equations.

a. \frac{dy}{dx} = ^{3} \sqrt{\frac{x}{y}} b. \frac{d^{2}x}{dt^{2}} = −ω^{2}x

c. \frac{d^{2}x}{dt^{2}} = −g

3. section 5.2, #22,35,36

4. For each function below, determine whether or not it is Riemann integrable on the interval (0,1], and fully explain your result: (a) f(x) = sin (1/x), (b) f(x) = 1/\sqrt{x^{2} sin x}, (c) f(x) = x for x rational and = −x for x irrational.

5. Calculate ∫_{1}^{2} (3x^{2} − 2) dx from the definition of the integral, that is, using Riemann sums.

6. Calculate the following integrals:

(a) ∫_{0}^{4} \sqrt{x} dx (b) ∫_{0}^{π/2} sin x dx (c) ∫_{1}^{3} \frac{1 − 3x^{3}}{x^{2}} dx

(d) ∫_{0}^{π} sin^{2} x dx (e) ∫_{0}^{π} x cos(x^{2} + π) dx (f) ∫_{−3}^{3} x^{3} dx

7. Find \frac{d}{dx} ∫_{0}^{x^{2}} tan θ dθ.